Eppsteinʹs bound on intersecting triangles revisited

Let S be a set of n points in the plane, and let T be a set of m triangles with vertices in S. Then there exists a point in the plane contained in Ω ( m 3 / ( n 6 log 2 n ) ) triangles of T. Eppstein [D. Eppstein, Improved bounds for intersecting triangles and halving planes, J. Combin. Theory Ser. A 62 (1993) 176–182] gave a proof of this claim, but there is a problem with his proof. Here we provide a correct proof by slightly modifying Eppsteinʹs argument.